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A circle has equation x^{2}-8x+y^{2}-6y=d. A line is tangent to this circle and passes through points A and B, (0,17) and (17,0) respectively. Find the radius of the circle.

Gradient of line: (0-17)/(17-0)= -1 equation of line: y-y 1 =m(x-x 1 ) y-17=-1(x-0) y=17-x equation of circle: (x-4) 2 +(y-3) 2 -25=d (completing the square) (x-4) 2 +(y-3) 2 =d+25 Substitute equation of lin...
AH
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Using differentiation, show that f(x) = 2x^3 - 12x^2 + 25x - 11 is an increasing function.

First compute the derivative of f(x) using the power rule on each term. f(x) = 2x^3 - 12x^2 + 25x - 11 so f'(x) = 6x^2 - 24x + 25. Now complete the square for the derivative. f'(x) = 6 * ((x-2)^2 - 4) + 25 =...
MT
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f(x) = 3x^3 – x^2 – 20x – 12 (a) Use the factor theorem to show that (3x + 2) is a factor of f(x). [2 marks] (b) Factorise f(x) fully. [3 marks]

(a) Factor theorem hence, use x = -2/3. Sub in : 3(-2/3) 3 - (-2/3) 2 -20 (-2/3) -12 = 0 (b) (3x+2)(ax 2 + bx+c) = 3x 3 – x 2 – 20x – 12 3ax 3 + (3b+2a)x 2 + (3c+2b)x + 2c = 3x 3 – x 2 – 20x – 12 Therefore, ...
DW
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Find the coordinates of the minimum point of the function y=(x-5)(2x-2)

At the minimum point the gradient is zero so dy/dx=0. To find dy/dx, first expand out the brackets so y=2x^2 - 12x + 10. Using differentiation dy/dx=4x - 12. At the minimum 4x-12=0 so 4x=12 therefore x=3. Pu...
PC
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The equation of the line L1 is y = 3x – 2 The equation of the line L2 is 3y – 9x + 5 = 0 Show that these two lines are parallel.

Two lines are parallel if they have the same gradient. This can be found by looking at the coefficient of x. When the equation is written in the form y=ax+b, with b a constant, the gradient of the line would...
JS
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